YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
E. Why call a set of ordered pairs a relation?
Numerical Exercise
- Plot the points (4,-4), (0,0) and (4,4). Join adjacent points by a straight line segments to obtain a set S of ordered pairs which does not satisfies the vertical line property. Use interval notation to describe the domain and range of this set. Give formula y = ?? for the line segment joining (0,0) and (4,4). Give another formula y = ??? for the line segment joining (0,0) and (4, -4). If (x,y) belongs to S and x = 3, what are the possible values of y? If (x,y) belongs to S and 0 < x < 4, what are the possible values of y?
Question 1 provides motivation for the set-based codification and definition of a relation S as a set of order pairs. The condition (x,y) belongs to a set S relates y to x by requiring y have a value in the intersection of S with the vertical line with horizontal coordinate x. The set S could be given by the solution set of an equation involving and thus relating the values of x and y.
Example: If [x,y] satisfies the equation
x2+y2 = 1
then [x, y] lies on the unit circle.
In this example,
- when x = b between -1 and +1, there are two solutions y of the equation x2+y2 = 1 , namely y = sqrt(1-b2) and y = -sqrt(1-b2) and so two point [x,y] on the line x = a satisfy the equation,
- when x = 1, only one point [x,y] on the line x = 1 satisfies the equation, namely [x,y]= [1,0] or y =0
- when x = a < -1, no real value of y and hence point [x,y] on the line x = a satisfies the equation as sqrt(1-a2) calls for the square root of a negative number.
For any real value given to x, there are zero, one or two values of y satisfying the equation.
In the development of mathematics, an equation involving x and y linked or related values. We see that an equation in x and y or alternatively, the set S of solutions of the equation relate or link the values of x and y. The set viewpoint allows for more relations that provided by algebraic equations alone.
To learn more, see Functions and Sets, a chapter in the site Volume 2, Three Skills for Algebra, online in full with postscripts.
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelFNs & Dependency FN With Finite Sets FN Vertical Line Rule FN Infinite Domains FN Sets-Theory FN Interval Notation FN: Sets - Continued (FN) Sets & Relations I (FN) Relations & Sets FN Source Target Domain Range (FN) Injective or Not (FN) Sign & Zero Analysis (FN) Increasing/Decreasing (FN) Extrema FN Numerical Exercises FN Step Sawtooth Abs.Value FN Horizontal Line Rule FN Inverse Functions FN Many Ways to Define
Area Entrance (C) Complex Numbers (FN) What are Functions? (FN) Functions - More SZM: Sign Analysis (L) Lines Summary (P) Polynomials (*,+,-) (Q) Quadratics (D) Simplify Square Roots (T) Unit Circle Trig Conic Sections Links More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!